Theorem aaanv 41895

Description: Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2332. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
aaanv	⊢ ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥∀𝑦(𝜑 ∧ 𝜓))

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem aaanv

Step	Hyp	Ref	Expression
1		nfv 1918	. . 3 ⊢ Ⅎ𝑦𝜑
2		nfv 1918	. . 3 ⊢ Ⅎ𝑥𝜓
3	1, 2	aaan 2332	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
4	3	bicomi 223	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥∀𝑦(𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 205   ∧ wa 395  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-nf 1788

This theorem is referenced by: (None)